The temperature should be <10°.The issue is, that we couldn't insure Fifo management of solder paste (Locking system inside the refrigerator , it could be like automatic dispenser)with normal domestic refrigerator. Any suggestions?

You can try by designing your own fixture to ensure the FiFo, but it will always depend on the operator to fill it in the righ order, I used to have a fixture like the one in the image, but it was installed next to the machine, not inside the fridge... just an idea...